This is for a problem of a medium through which light passes in 0 < y < h and we have that 0 < k < (1/h).

So we could do standard Euler-Lagrange and get , however I don't see how to solve this. If we note that there is no x-dependence then we can use that and get that however this does not seem good either. I am looking to show that the solution, y(x) is a parabola.